Airport Noise Regulation, Airline Service Quality,and Social Welfare

by

Jan K. Brueckner

Department of Economics

University of California, Irvine

3151 Social Science Plaza

Irvine, CA 92697

e-mail: jkbrueck@uci.edu

and

Raquel Girvin

Program in Transportation Science

University of California, Irvine

Irvine, CA 92697

e-mail: rgirvin@uci.edu

September 2006

Abstract

This paper explores the impact of airport noise regulation on airline service quality and airfares.It also characterizes the socially optimal stringency of noise limits, taking both noise damageand the various costs borne by airlines and their passengers into account. The analysis alsoinvestigates the effect of noise taxes, as well as the optimal level level of such taxes. Alongwith the companion paper by Girvin (2006a), this work represents the first complete theoreticalinvestigation into the economics of airport noise regulation using a model where the interestsof the key relevant stakeholders are captured.

Airport Noise Regulation, Airline Service Quality,and Social Welfare

by

Jan K. Brueckner and Raquel Girvin

1. Introduction

Despite dramatic reductions over the years in the noise produced by individual aircraft, air-

port noise remains a critical public policy issue today. Moreover, given the expected increases

in airline traffic and airport operations over the next decades, the noise issue will continue to

be a source of dissension. The nature of the problem is evident in the ongoing controversy

surrounding the planned expansion and reorientation of Chicago’s O’Hare airport. While the

expansion requires demolition of several hundred properties, it also reorients the airport’s flight

paths, so that a new set of households will be exposed to noise (see McMillen (2004)). Both of

these anticipated effects have led to vociferous opposition to the expansion from nearby resi-

dents, who have attempted to block the plan in court, despite its recent approval by the Federal

Aviation Administration (FAA). Similarly, in Orange County, California, concerns about noise

exposure blocked the construction of a new international airport on a decommissioned military

airbase, even with a shortage of airport capacity in the region (Kranser (2002)). At nearby

John Wayne Airport, departing flights must practice a steep, high-power climb maneuver to

quickly gain altitude before passing over the high-income community of Newport Beach, and

noise concerns in that community continue to limit daily flight volume at the airport.1

The dramatic gains in aircraft “quietness” over the jet age, which have ironically accompa-

nied ongoing noise concerns, are illustrated by comparing noise from a recent-vintage Boeing

737-700 and a 1967-vintage 727-200. The newer aircraft produces only one-third as much per-

ceived takeoff noise as its predecessor, despite similar passenger capacities. Because of such

gains in quietness, the number of U.S. residents exposed to significant aircraft noise fell by a

factor of 16 between 1975 and 2000 despite a more than three-fold increase in airline traffic

over the period. However, even with such gains in noise abatement, Figure 1 shows sharply

1

growing trends in various airport noise limits, such as operational curfews, noise quotas, and

noise surcharges, in the U.S. Such measures are even more widespread in Europe, as discussed

by Girvin (2000c).

Noise restrictions are likely to have important impacts on airline service quality and airfares.

Service quality may fall as various operational limits restrict flight frequency, and the expense

of making aircraft quieter, which raises their purchase price and operating cost, may be passed

on in higher airfares. Despite these possible linkages, the airline economics literature contains

no comprehensive analysis of the effect of noise regulation on airline service quality and fares.2

Because of this absence, no proper analysis of optimal noise regulations, which maximize social

welfare taking into account impacts on airlines and their passengers as well as noise victims,

has been possible. The purpose of this paper is to provide the missing analysis through the

use of a highly stylized, but suggestive, theoretical model.

The analysis draws on the scheduling model of Brueckner (2004), where higher flight

frequency benefits passengers by reducing “schedule delay” (allowing departures at more-

convenient times) while generating higher total noise. Noise per aircraft, denoted n, can

be reduced at a cost, which rises with aircraft size given that quieting a larger plane is more

expensive. The airline is viewed as choosing both n and aircraft size, along with flight fre-

quency and fares, to maximize profit subject to noise regulations. The manufacturer responds

to the resulting demand for aircraft quietness in its design decisions.

The analysis considers two different regulatory regimes involving explicit noise constraints,

along with an alternative regime where airlines pay noise taxes. The first type of noise con-

straint imposes a direct limit on noise per aircraft, with the constraint written as n ≤ n, where

n is the limit. Note that, under this constraint, n is removed as a choice variable for the airline.

The n limit is analogous to the FAA noise certification standard, which governs quietness levels

for new aircraft while also requiring retrofitting of older, noisier planes.3

The second type of constraint is a cumulative noise limit at an airport, which is written

nf ≤ L, where f is flight frequency (the number of flights) and L is total allowed noise for

each airline. Note that an airline has flexibility in meeting a cumulative limit because total

noise depends on both n and flight frequency. This type of constraint, among other noise

2

restrictions, is imposed by Amsterdam’s Schiphol Airport (AMS), London’s Heathrow Airport

(LHR), and Long Beach Airport (LGB) in California.4 Even though cumulative noise limits

are not widespread currently, Figure 1 suggests that they are likely to become more common

as time passes, making an understanding of their effects important.

Under noise taxation, explicit constraints are removed, but the airline instead pays a tax of

t per unit of noise, so that its total tax liability is tnf . Although noise taxes are not used much

in the U.S., they are more common elsewhere, being levied through landing-fee adjustments

that depend on an aircraft’s noise level. For institutional background, see Nero and Black

(2000) and Morell and Lu (2000).

In the analysis, the effect of each of the three regulatory regimes is considered in isolation,

ignoring the fact that different regimes often coexist in reality. This potentially unrealistic

approach is meant to gain insight into the economics of the regimes, and its practical lessons

are considered in the conclusion. Another unrealistic element is the linear fashion in which

noise is added across flights under the cumulative limit. Under the actual regulations, noise

is added in a semi-logarithmic fashion, which makes total noise more sensitive to noise per

flight than to the number of flights.5 While incorporating this feature would complicate the

analysis, the main qualitative conclusions are likely to be unaffected.

The analysis solves the airline’s profit-maximization problem under cumulative and per-

aircraft noise regulation and under the noise-tax regime. Comparative-static analysis shows

the effects of parameter changes on the airline’s choice variables, results that are immediate

given that the model generates closed form solutions for all the variables. In addition, a key

equivance result between noise taxation and cumulative noise regulation is established.

While the cumulative or per-aircraft noise limits are treated as parameters in the

comparative-static analysis, the next step is to consider the social planner’s problem, where L or

n is chosen optimally, taking into account the environmental damage from noise. Even though

the choice of a regulatory regime (cumulative vs. per-aircraft noise regulation) is immaterial

in characterizing the first-best social optimum, the two regimes generate different outcomes in

the second-best case, where the planner must accept the (inefficient) profit-maximizing choices

of the airline. An interesting question is then which regime yields a higher welfare level. This

3

question cannot be addressed analytically, but it is investigated via numerical analysis. The

analysis also characterizes the noise tax rates that support the first-best and second-best social

optima.

The entire analytical agenda described above can be carried out either under a monopoly

market structure, with a single airline, or in an oligopoly setting. Since the monopoly case

has already been analyzed by Girvin (2006a), the present paper analyzes an oligopoly model,

with the specific case of a duopoly considered. As seen below, the duopoly setup is actually a

special case of the monopoly model, and the results for this latter model are briefly summarized.

The duopoly model is drawn from the analysis of Brueckner and Flores-Fillol (2006), and its

key element is idiosyncratic airline brand loyalty on the part of consumers, which governs

competition between the two duopoly carriers.

The plan of the paper is as follows. Section 2 develops the basic model. Section 3 introduces

the cumulative and per-aircraft noise constraints and solves the airline profit-maximization

problem under these two different regimes. Section 4 analyzes noise taxation, section 5 carries

out the welfare analysis, and section 6 offers conclusions.

2. The Model

To start the analysis, it is useful to focus on airline costs. Following Brueckner (2004), the

cost of each flight, exclusive of noise-abatement cost, is given by θ + τs, where s equals seats

per flight. Each flight thus entails a fixed cost θ as well as a variable cost τ per seat. While

this specification may not be completely realistic, it captures the fact that an airline’s cost per

seat (given by θ/s + τ ) falls as aircraft size increases. Multiplying by flight frequency yields a

total cost expression, f(θ + τs).

Noise-abatement cost per flight, which is embodied in the aircraft design, is assumed to

depend on both aircraft size and quietness, being written g(s, n). The function g is increasing

in s, indicating that it costs more to achieve a given level of quietness for a larger aircraft. In

addition, g is decreasing in n, noise per aircraft, indicating that a quieter aircraft of a given

size generates higher costs, partly through lower fuel efficiency. Since noise abatement affects

both the purchase price and operating cost of an aircraft, g should be viewed as giving the

4

annualized abatement cost.

The function g is assumed to be homogeneous of degree zero, so that proportional increases

in aircraft size and noisiness leave abatement costs unchanged. A particular functional form

is then imposed for analytical tractability, with g(s, n) ≡ εs/n, where ε > 0 is the abatement-

cost parameter. This form embodies the above assumptions on g’s partial derivatives, and its

simplicity yields closed-form solutions for the variables of interest. With total noise abatement

cost equal to fεs/n, the airline’s full cost is written

f(θ + τs) + εfs/n = θf + (τ + ε/n)fs. (1)

Turning to the demand side of the model, the analysis focuses on a single city-pair market

served by two airlines, although the extension to multiple markets is straightforward. Utility for

a consumer traveling by air between the cities is given by c+travel benefit−schedule delay cost,

where c is consumption expenditure and schedule delay measures the difference between the ac-

tual and desired departure times. To represent schedule delay and its cost, suppose consumers

must commit to travel before knowing their preferred departures times, which are uniformly

distributed around a circle. Letting T denote the time circumference of the circle, consumer

utility from flying on airline 1 then depends on expected schedule delay, which equals T/4f1,

where f1 is number of (evenly spaced) flights operated by the airline (T/4f2 is the analogous

expression for airline 2). These expressions give the expected interval between the desired

departure time and the nearest flight time. Schedule delay cost is then given by a disutility

parameter δ > 0 times the above expressions, thus equaling δT/4fi ≡ γ/fi for airline i, i = 1, 2,

where γ = δT/4.

Travel benefit has two components: b, equal to the gain from air travel, and a, an airline

brand-loyalty variable, which gives the additional gain from using airline 1 (relative to travel

on airline 2). Brand loyalty varies across consumers, with a uniformly distributed over the

interval [−α/2, α/2], where α > 0, so that half the consumers prefer airline 1 and half prefer

airline 2.

For consumers using airline i, consumption expenditure equals y − pi, where y is income,

assumed to be uniform across consumers without loss of generality, and pi is airline i’s fare.

5

Utility from travel on airline 1 is thus given by y − p1 + b + a − γ/f1, and utility from travel

on airline 2 is given by the analogous expression with a = 0. Assuming that the consumer

undertakes air travel, his choice is between the two airlines, and airline 1 will be chosen when

y − p1 + b + a − γ/f1 > y − p2 + b − γ/f2, (2)

or when

a > p1 − p2 + γ/f1 − γ/f2. (3)

Thus, for the consumer to choose airline 1, brand loyalty to 1 must be at least as large as the

fare plus delay-cost difference between airlines 1 and 2. Otherwise, conditional on flying, the

consumer will choose airline 2.

To compute airline 1’s traffic using (3), let the consumer population size be normalized to

unity. Then, recalling that a is uniformly distributed over the range [−α/2, α/2], airline 1’s

traffic is given by

q1 =

∫ α/2

p1−p2+γ/f1−γ/f2

1

αda, (4)

where 1/α gives the density of a. Carrying out the integration in (4) yields

q1 =1

2− 1

α[p1 − p2 + γ/f1 − γ/f2] , (5)

with q2 given by the analogous expression with the 1 and 2 subscripts interchanged. Note from

(5) that airline 1 loses traffic when its fare rises or its frequency falls, while it gains traffic when

p2 rises or f2 falls.6

The monopoly model, analyzed by Girvin (2006a), is generated by considering an “outside

option” not involving air travel, which could represent use of a different transport mode or not

traveling at all. It yields benefits e net of any fare paid or schedule delay incurred and thus a

utility level of y+e. The monopoly model emerges when a consumer who is indifferent between

the airlines, with a = 0, prefers the outside option to air travel. Normalizing e to zero, this

6

outcome requires y− p1 + b− γ/f1 < y, or b < p1 + γ/f1, and b < p2 + γ/f2. While consumers

with a values near zero will not fly when these equalities hold, stronger airline brand loyalty

can induce air travel. For example, a passenger loyal to airline 1 will find air travel and the

outside option equally attractive when his loyalty value satisfies y − p1 + b + a − γ/f1 = y, or

a = p1 + γ/f1 − b.7 Thus, consumers with a values larger than

a = p1 + γ/f1 − b (6)

will then strictly prefer travel on airline 1 to the outside option, so that airline 1’s traffic is

equal to8

q1 =

∫ α/2

a

1

αda =

1

2− p1 + γ/f1 − b

α. (7)

Since airline 2’s fare and frequency choices do not appear in (7), competition between the

carriers is not present, with airline 1 effectively becoming a monopolist for passengers with

positive a values (airline 2 becomes a monopolist for negative-a passengers).

For the duopoly case to be relevant, b ≥ p + γ/f must hold at the p and f values that

emerge in a symmetric duopoly equilibrium. Otherwise, passengers with a values near zero

will prefer the outside option, making the monopoly case relevant instead. Accordingly, b

is assumed to be large enough to ensure satisfaction of the above inequality, with the exact

lower bound discussed below.9 Results for the monopoly case, which are presented by Girvin

(2006a), are summarized once the duopoly analysis is complete.10

It should be noted that, while a monopoly airline’s equilibrium traffic level depends on its

fare and frequency, as can be seen from (7), each duopoly airline carries an equilibrium traffic

level of 1/2 in a symmetric equilibrium regardless of its choices. From (5), q1 = 1/2 holds when

fares and frequencies are the same across carriers. In reaching the equilibrium, however, each

carrier takes into account traffic gains and losses to the other carrier as it adjusts its choice

variables, as seen below. But since no consumer is assumed to choose the outside option,

the carriers battle for a fixed amount of total traffic, which is split evenly between them in

equilibrium. This feature of the model accounts the simplicity of the ensuing analysis and the

7

emergence of clear-cut comparative-static effects, which are mostly absent in the monopoly

case.

With the traffic solution in (5) relevant for the duopoly case, it can be combined with the

previous cost expression in (1) to yield profit. Multiplying (5) by p1 and subtracting (1) (with

subscripts added), airline 1’s profit is equal to

π1 = p1q1 − θf1 − (τ + ε/n1)f1s1

= p1q1 − θf1 − (τ + ε/n1)q1

= (p1 − τ−ε/n1)

(1

2− 1

α[p1 − p2 + γ/f1 − γ/f2]

)− θf1, (8)

Note that (8) uses the equality f1s1 = q1, which says that the number of flights times seats per

flight equals total traffic. Using (8), the subsequent analysis considers the profit-maximization

problem under the two different regulatory regimes.

3. Airline Choices Under Noise Constraints

3.1. The case of a cumulative constraint

Suppose that noise regulation takes the form of a cumulative noise constraint, which im-

poses a common limit on the total noise generated by each airline, being written nifi ≤ L,

i = 1, 2. Since this constraint will bind in equilibrium, it can be used to eliminate n1, which

can be written n1 = L/f1. Substituting in (8), the multiplicative p1 − τ − ε/n1 term becomes

p1 − τ − εf1/L. The resulting objective function then depends only on the choice variables p1

and f1.

In modeling airline decisions, an important question concerns the timing of choices. While

both flight frequencies and fares can be adjusted relatively easily, fares appear to be more

flexible than frequencies. As a result, under the most realistic approach, fares would be chosen

conditional on frequencies, which themselves would be set in a first stage, taking the second-

stage impact on fares into account (the outcome is a subgame perfect Nash equilibrium).

The analysis eventually adopts this approach, but it is useful to start by considering the less-

8

realistic case where fares and frequencies are chosen simultaneously in Nash fashion. Under this

approach, airline 1 simultaneously sets the levels of p1 and f1, taking p2 and f2 as parametric.

The first-order conditions for the simultaneous-choice problem are

∂π1

∂p1= q1 − p1 − τ − εf1/L

α= 0 (9)

∂π1

∂f1= − ε

L

(1

2− 1

α[p1 − p2 + γ/f1 − γ/f2]

)+

γ(p1 − τ − εf1/L)

αf21

− θ = 0. (10)

While the derivatives ∂2π1/∂p21 and ∂2π1/∂f2

1 are easily seen to be negative, the remaining

second-order condition (positivity of the Hessian matrix of π1) is assumed to hold.11

Given the symmetry of the setup, the equilibrium fares and frequencies will be symmetric

across carriers, with the values denoted p and f , and each airline’s traffic will equal 1/2.

Imposing the latter condition in (9) and rearranging, the fare can be written

p = α/2 + τ + εf/L. (11)

Thus, the fare equals the full marginal cost of a seat, given by the operating cost τ plus noise

abatement cost per seat (εf/L),12 plus a markup that depends on brand loyalty. Note that,

as the extent of brand loyalty rises, leading to a higher α/2, a carrier loses less traffic from

raising its fare, so that p increases.

After substituting (11) into (10) and imposing symmetry, the equation can be solved for

the equilibrium frequency, which equals

f =

√γ

2θ + ε/L. (12)

This solution shows that frequency is increasing in the disutility of schedule delay (γ) and

decreasing in the fixed cost per flight (θ), both natural conclusions. Since the fare in (11)

increases with frequency, p is then increasing in γ and decreasing in θ. Note that the last

9

conclusion, which says that p is inversely related to the fixed cost per flight, is counterintu-

itive. The explanation is that the reduction in service quality caused by the higher θ reduces

willingness to pay for air travel. Observe also that f is independent of the marginal seat cost

(τ ) and the extent of brand loyalty (α).13

The key comparative-static question, however, concerns the effect of the noise limit L on

the airline’s choice variables. When the noise constraint is relaxed by increasing L, (12) shows

that frequency rises, with the denominator expression becoming smaller. This conclusion is

natural since, as L rises, the airline can raise f for a given n without violating the constraint.

With L and ε appearing in ratio form in (12), the opposite conclusion applies to an increase

in ε: by raising the cost per flight, a higher noise-abatement cost lowers frequency.

Turning to the impact of L on p, the effect is not immediately apparent given that p has an

direct inverse dependence on L from (11) as well as an indirect dependence via f . However, after

substituting (12) into (11) and rearranging, the last term reduces to√

γε2/(2θL2 + εL), which

is decreasing in L. Therefore, relaxation of the noise constraint leads to a lower equilibrium

fare, a conclusion that shows how the cost of airline travel is affected along with service quality

(f) by the stringency of noise regulation. Since L and ε enter asymmetrically in the previous

expression, their impacts are no longer mirror images. But it is easily seen that the expression

is increasing in ε, so that the fare naturally rises with the cost of abatement.

Aircraft quietness and size, n and s, can be recovered from the f solution, allowing further

comparative-static effects to be derived. Since s = q/f = 1/2f , the parameter impacts on

aircraft size are opposite to those on flight frequency. With n = L/f , substitution of (12)

yields

n =

√2θL2 + εL

γ, (13)

so that n is increasing in L and ε, both natural conclusions. Thus, aircraft quietness falls (n

rises) as the noise constraint is relaxed or the cost of abatement increases. In addition, (13)

shows that a higher fixed cost per flight raises n, while an increase in the delay disutility has

the opposite effect. All of these comparative-static results are summarized in Table 1.

These results show that tighter noise regulation under a cumulative constraint hurts airline

10

passengers, with the impact operating through two channels. Flight frequency falls, reducing

service quality, and fares rise, so that the cost of travel escalates as it convenience declines.

For tighter regulation to be desirable, these losses must be offset by social gains from a lower

noise level, as analyzed below.14

3.2. Sequential choice under a cumulative constraint

Using the simultaneous-choice case as a benchmark, consider now the case where frequen-

cies and fares are chosen sequentially. Under sequential choice, fares are set conditional on

frequencies, with frequencies then chosen in a first stage, taking into account the second-stage

impact on fares.

To analyze this case, the first-order condition for choice of p1 (eq. (9)) is supplemented

with the analogous condition for airline 2, gotten by reversing the 1 and 2 subscripts in (9).

The equations are then solved for p1 and p2 as functions of f1 and f2. The solution for p1 is

given by

p1 = α/2 + τ − 1

3

(γ

f1− γ

f2+

2εf1 + εf2

L

), (14)

and the p2 solution is gotten by reversing the 1 and 2 subscripts. These solutions are substituted

into the profit function (8) for airline 1, which is differentiated with respect to f1 treating f2 as

parametric. Symmetry is imposed in the resulting first-order condition, which is then solved

for f .15 The result is

f =

√γ

3θ + ε/L, (15)

which differs from (12) by the presence of the factor 3, rather than 2, in the denominator.

Thus, for a given L, frequency is smaller in the sequential-choice case. The p solution in (11),

which remains relevant, shows that the smaller f leads to a lower fare under sequential choice.

In addition, the lower f leads to a smaller aircraft size, and aircraft are also noisier under

sequential choice (the 2 factor in (13) is replaced by 3).

Despite these differences, the timing of airline decisions has no effect on the main

comparative-static properties of the equilibrium. In particular, it is easy to see that the change

in the f solution leaves the signs in Table 1 unaffected, so that the table applies to both the

11

sequential and simultaneous-choice cases.16

The key conclusions of the preceding analysis are then summarized as follows:

Proposition 1. A reduction in the allowed noise level under a cumulative noiseconstraint leads to lower flight frequency, larger and quieter aircraft, and a higher fare,regardless of whether airline choices are simultaneous or sequential. Airline passengersare hurt by the tighter noise limit.

3.3. The case of a per-aircraft noise constraint

Suppose now that noise regulation takes the form of a per-aircraft noise constraint, written

as ni ≤ n, i = 1, 2. Since the constraint will bind, airline 1’s objective function is given by (8)

with n1 replaced by n. Under simultaneous choice, the previous first-order condition (9) for p1

remains relevant, but with L replaced by f1n. Imposing symmetry, the fare solution is then

p = α/2 + τ + ε/n, (16)

which corresponds to the previous solution in (11) after substitution for L. This equivalence

implies that, if L under the cumulative constraint is set so that noise per aircraft in (13) is

equal to n, the same fare levels emerge under the two regimes.

The first-order condition for f1 is now

∂π1

∂f1=

γ(p1 − τ − ε/n)

αf21

− θ = 0, (17)

and, substituting (16), it yields a symmetric frequency solution of

f =

√γ

2θ. (18)

Under sequential choice, which is analyzed by following the same steps as before, the fare

solution in (16) remains relevant but frequency is given by

f =

√γ

3θ, (19)

12

where a factor of 3 replaces the 2 in the simultaneous solution, as in the cumulative case.17

Under both the simultaneous and sequential solutions, frequency is increasing in γ and

decreasing in θ, as in the cumulative case. But frequency does not depend on n and is thus

independent of the stringency of the noise constraint, in contrast to the cumulative case. With

f independent of the noise limit, nf and thus total noise L rises as n is raised, and L rises

with γ and falls with θ, holding n fixed. In addition, since the fare solution in (16) does not

involve f , p is now independent of γ and θ, although the fare responds positively to α and

τ , as before. These comparative-static effects are summarized in Table 2. Airline passengers

are hurt, as in the cumulative case, by a tighter noise constraint, but the effect now operates

through a single channel, a higher fare, with service quality unchanged.18

As in the cumulative case, sequential choice yields a higher frequency and smaller aircraft

size than simultaneous choice. But since p is independent of f , fare levels are the same under

the two choice scenarios, in contrast to the previous conclusion.

A more interesting observation, however, is that frequency is always higher under a per-

aircraft noise constraint than in the cumulative case, with the opposite conclusion applying to

aircraft size. These results follow because the simultaneous frequency solution in (18) is larger

than the corresponding cumulative solution (12), with the same comparison applying to the

sequential solutions (19) and (15). Apparently, lower frequencies arise in the cumulative case

because a reduction in f provides a means of satisfying the cumulative constraint.

Summarizing the main conclusions of the above analysis yields

Proposition 2. (i) A reduction in the allowed noise level under a per-aircraft con-straint raises the fare, harming airline passengers, while having no effect on flightfrequency or aircraft size. Total noise falls.

(ii) Flight frequency is always higher and aircraft size lower under a per-aircraft noiseconstraint than under a cumulative constraint, but fares are identical in the two caseswhen the constraints are set to achieve the same noise per aircraft.

3.4. The monopoly case

In the monopoly case, analyzed by Girvin (2006a), different comparative-static results

emerge. Although closed-form solutions for the decision variables are not available, her analysis

13

shows that a tightening of the cumulative noise constraint reduces flight frequency, as in

Proposition 1. A lower L also affects traffic q, which no longer assumes the constant equilibrium

value of 1/2, causing it to drop. With f and q both falling, the change in aircraft size is

ambiguous, and the same conclusion applies to aircraft quietness and the fare. Under a per-

aircraft noise constraint, a reduction in n lowers f , in contrast to Proposition 2, while also

reducing traffic. In addition, aircraft size falls. These conclusions show that the duopoly and

monopoly equilibria have very different properties.

4. Noise Taxation

4.1. The equivalence of noise taxation and cumulative regulation

Under noise taxation, the airlines pay a tax of t per unit of noise.19 As a result, the

additional term tf1n1 is subtracted in the profit expression (8), which becomes

π1 = (p1 − τ − ε/n1)

(1

2− 1

α[p1 − p2 + γ/f1 − γ/f2]

)− θf1 − tf1n1. (20)

With simultaneous choice, this expression is maximized by choice of p1, f1, and n1 without the

presence of any constraint. The first-order condition for p1 reduces to the previous condition

(16), with n replaced by n1. After imposing symmetry, the first-order conditions for f1 and n1

reduce, respectively, to:

nf =ε

2tn(21)

f2 =γ

2θ + 2tn. (22)

These conditions determine the symmetric equilibrium values of f and n conditional on t.

To compare the outcome to the other regimes, it is useful to change variables so that (21) and

(22) are expressed in terms of f and L, where L is endogenous. Using (21) along with nf = L

to eliminate 2tn, (22) becomes

f =

√γ

2θ + ε/L, (23)

14

which is the same as the cumulative frequency solution (12) under simultaneous choice. In

addition, since (21) can be rewritten as (nf)2/f = ε/2t, the condition becomes

− εf

L2+ 2t = 0. (24)

Substituting (23) into (24) and simplifying yields the following condition that determines L as

a function of t: √γε

8θL4/ε + 4L3= t. (25)

To obtain the entire solution under noise taxation for a given t, (25) is solved for the corre-

sponding L, and substitution in (23) then yields f and consequently n. The fare solution in

(16) then yields p.

Because (23) is the same as the cumulative frequency solution, a key implication of (23)

and (25) is that noise taxation is equivalent to cumulative noise regulation. In other words, the

outcome generated by a particular cumulative noise limit L# can be exactly replicated under

noise taxation. To do so, L# is plugged into (25), and t is set at the resulting value, denoted

t#. When the airlines maximize facing t#, a total noise level of L# per carrier is generated

endogenously, and the f , n and p values corresponding to the cumulative limit then emerge.

The same conclusion emerges under sequential choice. Fares are again chosen in a second

stage, while airline 1 chooses f1 and n1 in the first stage taking the second-stage fare impacts

into account and viewing f2 and n2 as parametric. Given this behavior, a lengthy derivation

yields symmetric first-order conditions with the same form as (21) and (22). The difference

is that a factor of 3 replaces the 2 in each instance in these equations.20 As a result, 3

replaces 2 in (23) and (24), and 18 replaces 8 in (25). Since the modified (23) is then the

same as the sequential solution in (15), the same argument as before establishes that noise

taxation is equivalent to cumulative noise regulation under sequential choice. In other words,

the cumulative outcome for a given L = L# can be exactly replicated under noise taxation by

a suitable choice of t. Summarizing yields21

Proposition 3. Noise taxation and cumulative noise regulation are equivalent, re-gardless of whether airline choices are simultaneous or sequential. In particular, the

15

outcome under cumulative regulation can be exactly replicated by suitable choice of thenoise tax t.

The apparent explanation for this result is that both the tax and cumulative regimes

give the airline the same margins of adjustment (f and n), even though these variables must

satisfy a constraint in the cumulative case. In the per-aircraft regime, by contrast, the airline’s

adjustment margins are limited.

4.2. Comparative statics under noise taxation

Inspection of (25) shows how parameter changes affect L. First, since the LHS is decreasing

in L, if follows that a higher t reduces total noise per carrier, a natural result. In addition, it

is easily seen that L is decreasing in θ and increasing in both γ and ε, while being independent

of α and τ . Taking account of the effects on L, inspection of (23) shows that f is decreasing

in t and θ and increasing in γ, while an additional argument shows that f decreases with ε.

Since n = ε/2Lt from (21), eliminating t using (25) yields n =√

(2θL2 + εL)/γ. Using the

impacts on L, it follows from the latter solution that n is decreasing in t and increasing in ε.

Again using impacts on L, the first solution shows that n is decreasing in γ and increasing

in θ. Finally, using (16) with n in place of n, inspection shows that p is increasing in t and

decreasing in γ and θ, and an additional argument shows that p increases with ε.22

These results are summarized in Table 3. Note that the comparative-static effects of ε, τ ,

α, γ, and θ are naturally the same as in the cumulative case (compare the last five columns

of Tables 1 and 3). Finally, observe that since f falls and p rises with t, an increase in the

noise tax hurts airline passengers. These results, which also apply to the sequential case, are

summarized as follows

Proposition 4. An increase in the noise tax leads to less total noise, lower flightfrequency, larger and quieter aircraft, and a higher fare, regardless of whether airlinechoices are simultaneous or sequential. Airline passengers are hurt by the higher noisetax.

16

5. Welfare Analysis

This section of the paper carries out welfare analysis.23 Social welfare in the model is equal

to consumer surplus plus airline profit minus environmental noise damage. Since the fare paid

is just a transfer between consumers and the airlines, while the benefits from travel and brand

loyalty are constants, the planner can focus on a welfare measure equal to airline costs plus

the cost of consumer schedule delay plus noise damage. Imposing symmetry on the airlines,

and writing welfare in terms of L, the welfare expression equals

W = τ + 2θf +εf

L+

γ

f+ λ2L, (26)

where noise damage is assumed for simplicity to be a linear function of total noise (equal to

2L), with parameter λ.24 To interpret the rest of (26), recall that total traffic is equal to unity,

which then multiplies the marginal seat cost τ and schedule delay cost, γ/f . With two airlines,

fixed cost acquires a factor of 2, and total noise abatement cost is 2f(εs/n) = ε/n = εf/L,

where the first equality uses the fact that 2fs equals the total unitary traffic level. The

planner’s goal is to minimize (26) by choice of f and L. Equivalently, (26) could be rewritten

in terms of n, which would be chosen optimally along with f .25

Note that, because the fare cancels in computing W , the impacts of noise regulation on

passengers that operate through p are irrelevant in the welfare analysis. Only those effects

operating through schedule-delay costs matter.

5.1. The first-best optimum and the second-best optimum under a cumulative constraint

Viewing L as the choice variable, it is useful to derive the social optimum in two steps.

First, f is chosen to minimize (26) conditional on L, yielding the first-order condition

2θ +ε

L− γ

f2= 0, (27)

and the resulting frequency solution

f∗(L) ≡√

γ

2θ + ε/L, (28)

17

where f∗(L) denotes the optimal frequency conditional on L. Next, (28) is substituted back

into (26), which is differentiated with respect to L. Using the envelope theorem, the resulting

first-order condition is

−εf∗(L)

L2+ 2λ = 0, (29)

which says that the extra noise damage should equal the saving in abatement cost when L is

marginally increased. After substituting (28) into (29) and simplifying, the socially optimal

noise level, denoted L∗ and referred to as the first-best optimal value, turns out to be the root

of a 4th degree polynomial in L. The first-best optimal f then equals f∗(L∗), and the optimal

n is given by n∗ = L∗/f∗(L∗).

The conditionally optimal f solution in (28) reveals the efficiency of airline choices under

a cumulative constraint, for a given value of L. Since (28) and (12) are the same, it follows

that for a given L, the airline’s frequency choice is efficient, provided that its choices are made

in simultaneous fashion. However, if choice is sequential, then the airline’s chosen frequency

is lower than the one desired by the planner, as seen by comparing (15) and (28), and aircraft

size is correspondingly too large.

With the airline’s choices efficient under simultaneous choice, it follows that the planner

could set L at L∗, allow the airline to maximize profit, and be assured that the outcome would

be socially optimal. However, under more-realistic sequential behavior, the airline makes the

wrong choices conditional on L, which means that setting L at L∗ would fail to generate the

social optimum. Thus, in the sequential case, the planner must take the airline’s distorted

choices into account in setting L, which leads to the second-best optimal level under sequential

choice, denoted L.

To characterize L, let f (L) denote the airline’s profit-maximizing sequential frequency

choice conditional on L, given by (15). With the planner obliged to accept airline choices, his

objective function comes from substituting f(L) into (26), which yields

W = τ + 2θf (L) +εf(L)

L+

γ

f(L)+ λ2L. (30)

18

Differentiating (30), L must satisfy

−εf(L)

L2−(

γ

f (L)2− 2θ − ε

L

)f ′(L) + 2λ = 0. (31)

Since the term in parentheses in (31) reduces to θ using (15), the condition can be written

−εf(L)

L2− θf ′(L) + 2λ = 0. (32)

Substituting f(L) and simplifying, (32) reduces to a 6th degree polynomial equation in L,

which determines L, the second-best optimal value.

An interesting question is whether L is larger or smaller than the first-best optimal value

L∗. In other words, when the planner must accept the airline’s inefficient choices, does he set a

cumulative noise limit looser or tighter than the first-best level? To answer this question, (32)

must be compared to the first-best condition in (29). As functions of L, both LHS expressions

must be upward-sloping curves at the respective optima to ensure minimization of W , which

means that the higher curve is associated with the smaller L solution. On the one hand, the

presence of the term −θf ′(L), which is negative by Proposition 1, tends to make the second-

best curve from (32) lower than the first-best curve, which in turn tends to make L larger

than the L∗. The intuitive explanation comes from recalling that the conditional sequential

frequency solution is inefficiently low, with f(L) < f∗(L). Given f ′(L) > 0, it follows that

the planner can use his second-best L choice to raise f from its inefficiently low level, an

intervention that would require setting L above its first-best value. On the other hand, the

fact that f(L) < f∗(L) makes the initial term in the second-best expression from (32) larger

(less negative) than the corresponding term in the above first-best expression, which tends to

make the second-best curve higher than the first-best curve.

Despite this apparent indeterminacy, once the f∗ and f functions are substituted into (29)

and (32) and appropriate manipulations are made, it can be shown that L < L∗ holds (see

the appendix). This conclusion means that the latter effect discussed above (which depresses

L) dominates the goal of using L to boost the airlines’ inefficiently low flight frequency. With

19

L < L∗, it follows from (15) and (28) that f(L) < f∗(L∗), so that the second-best flight

frequency under sequential choice is inefficiently low. In addition, using L < L∗, it can be

shown that n∗ < n = L/f (L), where n is the second-best optimal n value (see the appendix).

Summarizing yields26

Proposition 5. The second-best cumulative noise limit under sequential choice isless than the first-best limit (L < L∗). In addition, the associated second-best flightfrequency is lower than the first-best level, and the second-best per-aircraft noise levelis higher than the first-best level.

5.2. What noise-tax levels support the optima?

Since noise taxation and cumulative noise regulation are equivalent, it follows that the

above optima can be generated by suitable selection of the noise tax t. Consider first the

case where airlines make simultaneous choices. As seen above, when L is set at L∗ under

cumulative regulation, airline decisions yield the first-best optimum. But since any outcome

under cumulative regulation can be generated by an appropriate noise tax, it follows that a

particular value of the tax yields the first-best optimum.

To derive that value, observe that (23) and (24) jointly determine the solutions for f and L

under a particular level of the noise tax t. But this equation system has the same form as (28)

and (29), which determine the first-best optimum. The two systems can be made identical,

ensuring that the noise-tax solution coincides with the social optimum, when t is set at the

appropriate value, denoted t∗. As can be seen, the correct value is t∗ = λ, indicating that the

noise tax should be set equal to the marginal noise damage, a natural conclusion.

Suppose instead that airline choices are sequential, so that the second-best optimum is

relevant. To find the noise tax that generates this optimum, recall that the solution under noise

taxation is given by (15) along with the modified version of (24), which is written −εf/L2+3t =

0. Thus, the equation determining L under noise taxation can be written −εf(L)/L2 +3t = 0.

For the resulting solution to coincide with the second-best optimum, t must be set at the value

t satisfying

−εf(L)

L2+ 3t = 0. (33)

20

But since (32) must hold at L = L, (33) can be used to eliminate the first term, allowing the

equation to be rewritten as

−3t − θf ′(L) + 2λ = 0. (34)

Rearranging then yields

t =2

3λ − 1

3θf ′(L) < λ ≡ t∗. (35)

(note that (32) implies positivity of t). Thus, the noise tax rate that generates the second-best

optimum under cumulative regulation is less than the marginal noise damage and thus less

than t∗. Note that this conclusion might seem inconsistent with the inequality L < L∗ from

Proposition 5 given that a low tax should be associated with a large, not a small, L. But since

this inverse relationship holds airline choice behavior fixed, while t and t∗ apply under different

behaviors (simultaneous vs. sequential choice), no inconsistency arises.

It is important to recognize that, since noise taxation and cumulative noise regulation are

equivalent under sequential choice, and since L is the best possible value of L, a tax rate

different from t (one associated with a different value of L) cannot yield a higher welfare level.

Thus, t is the welfare maximizing noise tax rate under sequential choice, and it is less than the

marginal noise damage λ. Summarizing yields

Proposition 6. If airline choices are simultaneous, the first-best optimum can begenerated by a noise tax equal to the marginal noise damage λ. If airline choices areinstead sequential, the welfare maximizing tax rate is less than λ, and it generates thesecond-best optimum under cumulative regulation.

5.3. Could per-aircraft noise regulation be superior?

When airline choices are simultaneous, per-aircraft regulation can never be superior to

cumulative regulation (or equivalent noise taxation) since these regimes can generate the first-

best optimum. Per-aircraft regulation is, by contrast, inefficient given that the simultaneous

frequency solution in (18) is always larger than the first-best optimal level, f∗(L∗). But when

choices are sequential, this conclusion can be overturned. To understand this claim, consider

21

the following inequalities, where f denotes the sequential frequency solution from (19) under

per-aircraft regulation:

f ≡√

γ

3θ>

√γ

3θ + ε/L≡ f(L) < f∗(L∗) (36)

Thus, f is larger than f(L), the second-best optimal frequency under cumulative regulation,

which in turn is smaller than the first-best optimal frequency f∗(L∗) by Proposition 5. Given

(36), f could be closer to f∗(L∗) than is f(L), raising the possibility that per-aircraft regulation

is superior to cumulative regulation.

Since welfare under the per-aircraft regime also depends on the magnitude of n, exploring

this question requires computation of the second-best n under sequential choice. To do so,

the welfare expression in (26) is rewritten in terms of n and f is then substituted, so that the

last four terms become 2θf + ε/n + γ/f + λ2nf . The second-best optimal per-aircraft noise

limit is found by differentiating this expression with respect to n, noting that f from (36) is

independent of n. The resulting first-order condition is

− ε

n2+ 2λf = 0. (37)

After using (36) to substitute for f , (37) can be solved for n and the result substituted into

the above welfare expression. If per-aircraft regulation is superior, the resulting welfare level

is higher than the one achieved under cumulative regulation.

This comparison cannot be carried out analytically, but illustrative numerical examples

can be computed. In generating the examples, the marginal-cost parameter τ is normalized

to one, while four other parameters are allowed to vary.27 The delay-disutility and fixed-cost

parameters γ and θ vary between 0.1 and 10.1 in steps of 1, while three combinations of

the abatement-cost and noise damage parameters are considered: (ε, λ) = (1, 1), (10, 1), and

(10,10). Since the model is highly stylized, realistic calibration is not possible, making these

parameter choices arbitrary. For each parameter combination, the polynomial equations for

the various optima have only one positive solution, and the curves are upward sloping at the

solution values, ensuring that these values represent minima for W .

22

The welfare comparison between the two regulatory regimes, given as ratios of the cumu-

lative and per-aircraft welfare levels, is shown in Table 4. The results show that welfare under

the per-aircraft regime tends to be larger than under the cumulative regime when γ and θ

are big and smaller otherwise, with the margin of reversal shifting outward as ε and λ assume

larger values. Note that in the third panel of Table 4, welfare is higher under the cumulative

regime for all the values of γ and θ. Summarizing yields

Proposition 7. When airline choices are sequential, the second-best welfare level maybe higher under per-aircraft noise regulation than under cumulative regulation or theequivalent tax regime.

Despite this conclusion, further inspection of Table 4 reveals that the choice between the

two regulatory regimes is actually a matter of indifference for most parameter values, with the

ratios usually very close to 1. However, when γ and θ are both small, the per-aircraft regime

becomes markedly inferior to the cumulative regime. With the regimes otherwise close to

equivalent, this finding suggests that cumulative noise regulation (or equivalent noise taxation)

may be preferable to per-aircraft regulation. Of course, this conclusion might be overturned

by results from other unexplored regions of the parameter space.

A further numerical observation concerns the magnitude of the welfare sacrifice from the

planner’s inability to implement the first-best optimum. Interestingly, this sacrifice is very

small: second-best welfare under the cumulative regime is never smaller than 98 percent of the

first-best level, and the per-aircraft regime yields a similarly good outcome over most of the

parameter region. But for those parameter values where the per-aircraft regime falls well short

of the cumulative regime, the loss it generates relative to the first-best optimum is similarly

large.28

6. Conclusion

This paper has explored the impact of airport noise regulation on airline service quality

and airfares. It has also characterized the socially optimal stringency of noise limits, taking

both noise damage and the various costs borne by airlines and their passengers into account,

while analyzing the use of noise taxes. Along with the companion paper by Girvin (2006a), this

23

work represents the first complete theoretical investigation into the economics of airport noise

regulation using a model where the interests of the key relevant stakeholders are captured.

The main conclusions of the analysis are as follows: (i) noise regulation harms airline pas-

sengers by raising fares and potentially reducing service quality; (ii) cumulative and per-aircraft

noise regulation have quite different effects on airline decisions; (iii) cumulative regulation ap-

pears to be superior from a social-welfare perspective; (iv) under realistic sequential airline

choice behavior, the best a planner can do under cumulative regulation is to use an ineffi-

ciently tight noise limit that yields lower-than-optimal flight frequency; (iv) noise taxation is

equivalent to cumulative noise regulation, generating exactly the same airline decisions when

the tax rate is suitably chosen.

The analysis has treated noise taxes and the two noise-regulation regimes, cumulative and

per-aircraft, as alternatives, when in fact these approaches often coexist in actual practice. In

other words, the FAA certification standard puts an upper bound on noise per aircraft, while

individual airports impose cumulative noise limits and other operating constraints, including

noise taxes. Given this real-world complexity, what practical lessons can be drawn from the

analysis? To reach an answer, the first observation is that most aircraft currently in production

already have lower noise signatures than the latest Stage 4 noise limits.29 This fact, along with

the presence of additional airport-level regulations, apparently means that airports do not

find the existing per-aircraft noise limits to be sufficiently stringent. Therefore, they impose

additional regulations, which generate a demand for aircraft quieter than the current FAA

standard.

Instead of allowing aircraft quietness to be effectively determined by airport-level policies,

the FAA could instead impose a per-aircraft limit that is even more stringent than the Stage

4 limit. But since a cumulative noise limit (or the equivalent tax) appears to be a superior

instrument, the analysis suggests that airport-level regulation may actually be a preferable

policy.

Although the highly stylized framework presented in this paper captures the essential

elements of the airport noise problem, additional refinements and extensions of the model

could be beneficial. For example, more-realistic specifications of the airline’s noise-abatement

24

cost function and the functions giving cumulative noise and noise damage would improve the

model’s accuracy. Moreover, adding the airport itself to the list of stakeholders in the model

could provide further insights. The airport, responding to political pressure to reduce noise,

could be portrayed as choosing the noise regulatory regime that maximizes its profit for a

given target noise level. In other extensions, concurrent implementation of noise-abatement

measures, such as a nighttime curfew and a noise tax, could be modeled, and airline asymmetry

(with one large and one small carrier) could be introduced. Analysis of these extended models

might, in some cases, require numerical methods. But the insights gained from numerical

models, especially those that can be calibrated with empirical data, might prove valuable in

guiding airport noise policies.

25

Appendix

Proof of Proposition 5:

Combining (29) and (32), L∗ and L must satisfy

εf(L)

L2+ θf ′(L) =

εf∗(L∗)L∗2 (a1)

Substituting for the f∗ and f functions using (15) and (28), (a1) becomes

ε

L2

√γ

3θ + ε/L+

θε

2L2(3θ + ε/L)

√γ

3θ + ε/L=

ε

L∗2

√γ

2θ + ε/L∗ (a2)

Rearranging and squaring both sides, (a2) reduces to

2θL∗ + ε

3θL + ε

[7θL + 2ε

6θL + 2ε

]2

=

[L

L∗

]3

. (a3)

Suppose that L ≥ L∗ holds. Then the LHS of (a3) must exceed 1. But after rearranging, the

resulting inequality reduces to

L∗ ≥ L

[108θ3L2 + 59θ2εL + 8θε2

98θ3L2 + 56θ2εL + 8θε2

]. (a4)

Since the ratio term in (a4) exceeds 1, the inequality implies L∗ > L, which contradicts the

initial assumption that L ≥ L∗, implying that L < L∗ must hold instead.

To establish n∗ < n, note that (a1) can be rewritten as

ε

nL+ θf ′(L) =

ε

n∗L∗ , (a5)

which implies nL > n∗L∗. The conclusion follows given L < L∗.

26

Curfews

Noise Charges

Noise Level Limits

Operating QuotasChapter 3 Restrictions

Noise Abatement Procedures

Figure 1. Trends in Airport Noise Restrictions

(Source: http://www.boeing.com/commercial/noise/airports2005.pdf)

Table 1.

Comparative-Static Effects Under a Cumulative Constraint

L ε τ α γ θ

f + − 0 0 + −n + + 0 0 − +

p − + + + + −s − + 0 0 − +

Table 2.

Comparative-Static Effects Under a Per-Aircraft Constraint

n ε τ α γ θ

L + 0 0 0 + +

f 0 0 0 0 + +

p − + + + 0 0

s 0 0 0 0 − +

Table 3.

Comparative-Static Effects Under Noise Taxation

t ε τ α γ θ

L − + 0 0 + −f − − 0 0 + −n − + 0 0 − +

p + + + + + −s + + 0 0 − +

27

Table 4. Ratio of Second-Best Welfare under Cumulative Noise Limit to Welfare under Per-Aircraft Noise Limit

( ε = 1, λ = 1) θ

γ 0.1 1.1 2.1 3.1 4.1 5.1 6.1 7.1 8.1 9.1 10.1 0.1 1.3634 1.0276 1.0028 0.9954 0.9924 0.9911 0.9905 0.9903 0.9903 0.9904 0.99051.1 1.1983 0.9994 0.9915 0.9903 0.9904 0.9908 0.9912 0.9917 0.9921 0.9925 0.99282.1 1.1638 0.9957 0.9906 0.9903 0.9908 0.9913 0.9919 0.9924 0.9928 0.9932 0.99363.1 1.1449 0.9940 0.9904 0.9905 0.9911 0.9917 0.9923 0.9928 0.9933 0.9936 0.99404.1 1.1322 0.9930 0.9903 0.9907 0.9914 0.9920 0.9926 0.9931 0.9936 0.9939 0.99435.1 1.1228 0.9924 0.9903 0.9908 0.9916 0.9923 0.9929 0.9934 0.9938 0.9942 0.99456.1 1.1154 0.9920 0.9903 0.9910 0.9918 0.9925 0.9931 0.9936 0.9940 0.9944 0.99477.1 1.1093 0.9917 0.9903 0.9911 0.9919 0.9926 0.9932 0.9937 0.9942 0.9945 0.99488.1 1.1042 0.9914 0.9904 0.9912 0.9921 0.9928 0.9934 0.9939 0.9943 0.9947 0.99509.1 1.0999 0.9912 0.9904 0.9913 0.9922 0.9929 0.9935 0.9940 0.9944 0.9948 0.9951

10.1 1.0960 0.9910 0.9905 0.9914 0.9923 0.9930 0.9936 0.9941 0.9945 0.9949 0.9952

Ratio < 1 ( ε = 10, λ = 1)

θ γ 0.1 1.1 2.1 3.1 4.1 5.1 6.1 7.1 8.1 9.1 10.1

0.1 1.8666 1.1877 1.0992 1.0615 1.0407 1.0277 1.0189 1.0127 1.0081 1.0046 1.00201.1 1.5717 1.0831 1.0326 1.0138 1.0046 0.9994 0.9962 0.9942 0.9928 0.9919 0.99132.1 1.5047 1.0633 1.0213 1.0064 0.9994 0.9957 0.9935 0.9922 0.9913 0.9908 0.99053.1 1.4668 1.0528 1.0156 1.0029 0.9970 0.9940 0.9923 0.9914 0.9908 0.9905 0.99034.1 1.4407 1.0460 1.0120 1.0006 0.9956 0.9931 0.9917 0.9909 0.9905 0.9903 0.99035.1 1.4209 1.0410 1.0094 0.9991 0.9946 0.9924 0.9913 0.9907 0.9904 0.9903 0.99036.1 1.4051 1.0372 1.0075 0.9980 0.9939 0.9920 0.9910 0.9905 0.9903 0.9903 0.99037.1 1.3920 1.0341 1.0059 0.9971 0.9934 0.9917 0.9908 0.9904 0.9903 0.9903 0.99048.1 1.3808 1.0315 1.0047 0.9964 0.9930 0.9914 0.9907 0.9904 0.9903 0.9903 0.99049.1 1.3711 1.0293 1.0036 0.9958 0.9926 0.9912 0.9906 0.9903 0.9903 0.9904 0.9905

10.1 1.3626 1.0275 1.0027 0.9953 0.9924 0.9911 0.9905 0.9903 0.9903 0.9904 0.9906

( ε = 10, λ = 10) θ

γ 0.1 1.1 2.1 3.1 4.1 5.1 6.1 7.1 8.1 9.1 10.1 0.1 2.6722 1.5514 1.3697 1.2807 1.2258 1.1879 1.1599 1.1383 1.1211 1.1070 1.09531.1 2.2096 1.3330 1.2027 1.1423 1.1068 1.0832 1.0665 1.0540 1.0444 1.0367 1.03062.1 2.1014 1.2852 1.1677 1.1144 1.0835 1.0634 1.0492 1.0388 1.0309 1.0246 1.01963.1 2.0394 1.2585 1.1485 1.0993 1.0711 1.0529 1.0402 1.0310 1.0240 1.0185 1.01424.1 1.9963 1.2404 1.1356 1.0893 1.0630 1.0461 1.0344 1.0259 1.0195 1.0146 1.01075.1 1.9635 1.2267 1.1261 1.0819 1.0570 1.0411 1.0302 1.0223 1.0164 1.0118 1.00826.1 1.9371 1.2159 1.1185 1.0761 1.0523 1.0372 1.0269 1.0195 1.0140 1.0097 1.00647.1 1.9151 1.2070 1.1124 1.0714 1.0486 1.0342 1.0243 1.0173 1.0121 1.0081 1.00498.1 1.8962 1.1994 1.1072 1.0675 1.0454 1.0316 1.0222 1.0155 1.0105 1.0067 1.00379.1 1.8798 1.1929 1.1027 1.0641 1.0428 1.0294 1.0204 1.0139 1.0092 1.0055 1.0027

10.1 1.8652 1.1871 1.0988 1.0612 1.0405 1.0275 1.0188 1.0126 1.0080 1.0046 1.0019

References

Brueckner, J.K., 2004. Network structure and airline scheduling, Journal of IndustrialEconomics 52, 291-311.

Brueckner, J.K, Flores-Fillol, R., 2006. Airline schedule competition. Unpublishedpaper, University of California, Irvine.

City of Long Beach, Airport Noise Compatibility Ordinance, Chapter 16.43, http://www.longbeach.gov/apps/cityclerk/lbmc/title-16/chapter-16-43.htm [accessed May 2006].

Federal Aviation Administration (FAA), 2005. FAA Stage 4 Aircraft Noise Standards,Federal Register, 70-127, pp. 38741-50.

General Accounting Office (GAO), 2000. Report to the Ranking Democratic Member,Committee on Transportation and Infrastructure, House of Representatives, Aviation andthe Environment: Airport Operations and Future Growth Present Environmental Chal-lenges, http://www.gao.gov/archive/2000/rc00153.pdf, [accessed May 2006].

Girvin, R., 2006a. Airport noise regulation, airline service quality, and social welfare: Themonopoly case, Unpublished paper, Program in Transportation Science, UC Irvine.

Girvin, R., 2006b. Economic Analysis of Aircraft and Airport Noise Regulation, UnpublishedPh.D. dissertation, Program in Transportation Science, UC Irvine.

Girvin, R., 2006c. Aircraft noise abatement and mitigation strategies, Unpublished paper,Program in Transportation Science, UC Irvine.

Hsu, C.-I., Lin, P.-H., 2005. Performance assessment for airport noise charge policies andairline network adjustment response, Transportation Research Part D 10, 281-304.

Kranser, L., 2002. Chronology of the War over El Toro Airport, Internet for Activists,Writers Club Press, San Jose.

McMillen, D., 2004, Airport expansions and property values: The case of Chicago O’Hareairport, Journal of Urban Economics 55, 627-640.

Morrell, P., Lu, C., 2000. Aircraft noise social cost and charge mechanisms–A case studyof Amsterdam airport Schiphol, Transportation Research Part D 5, 305-320.

Morrison, S.A., 1987. The equity and efficiency of runway pricing, Journal of Public Eco-nomics 34, 45-60.

28

Morrison, S.A., Winston, C., Watson, T., 1999, Fundamental flaws of social regulation:The case of airplane noise, Journal of Law and Economics 42, 723-743.

Nelson, J.P., 2003. Meta-analysis of airport noise and hedonic property values: Problemsand prospects, Journal of Transport Economics and Policy 38 (Part 1), 1-22.

Nero, G., Black, J., 2000. A critical examination of airport noise mitigation scheme and anaircraft noise charge: The case of capacity expansion and externalities at Sydney (KingsfordSmith) airport, Transportation Research Part D 5, 433-461.

Nero, G., Black, J., 1998. Hub-and-spoke networks and the inclusion of environmentalcosts on pricing, Transportation Research Part D 3, 275-296.

Oum, T.H., Zhang, Y., 1990. Airport pricing : Congestion tolls, lumpy investment, andcost recovery, Journal of Public Economics 43, 353-374.

Schiphol Group, 2003. Sustainable Development at Amsterdam Airport Schiphol, http://www.schiphol.nl/media/portal/schiphol-group/pdf/pdf-files/verslag-vo-2003-uk-v1-m56577569830644545.pdf, [accessed May 2006].

Schipper, Y., Nijkamp, P., Rietveld, P., 1998. Why do aircraft noise value estimatesdiffer?, Journal of Air Transport Management 4, 117-24.

Schipper, Y., Nijkamp, P., Rietveld, P., 2003. Airline deregulation and external costs:A welfare analysis, Transportation Research Part B 37, 699-718.

Wit, R.C.N., Davidson, M.D., Dings, J.M.W., 2003. Meeting external costs in the avia-tion industry, Report to the Commission for Integrated Transport, CE, Delft, http://www.cfit.gov.uk/docs/2003/aec/research/pdf/aec.pdf, [accessed May 2006].

Zhang, A., Zhang, Y., 2003. Airport charges and capacity expansion: effects of concessionsand privatization, Journal of Urban Economics 53, 54-75.

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Footnotes

∗We thank Kurt Van Dender and Ricardo Flores-Fillol for helpful comments. Any errors,however, are our responsibility.

1The U.S. General Accounting Office (2000) identified noise as the greatest environmentalconcern at 29 out of the 50 busiest airports.

2Nero and Black (2000) provide an analysis of the effect of per-flight taxes on an airline’soperations and network structure, motivating such taxation as a noise-abatement measure.However, their model lacks the richness of the framework developed below. Schipper, Ri-etveld and Nijkamp (2003) also consider the noise effects of airline deregulation in a modelwith frequency competition, but they do not analyze noise-abatement policies. Hsu and Lin(2005) analyze noise taxation in a complex simulation-based network model.

3The FAA recently adopted a new Stage 4 aircraft noise certification standard, which rep-resents a 10 decibel cumulative noise reduction below the current Stage 3 standard (FAA(2005)). Glasgow, Leeds, and Heathrow airports in the U.K. also set their own per-aircraftnoise limits and fine operators that exceed the limits.

4For each planning year, AMS establishes an annual total noise volume, which determinesthe number of available slots for airline operations, based on the noise characteristics offorecast fleet composition. If actual airline operations exceed the planned noise limit, thelaw requires the airport to curtail operations (Schiphol Group (2003)). LGB imposes acumulative noise exposure limit on passenger and cargo airlines. The number of permissibleairline operations at LGB depends on the cumulative noise from operations in the previousyear. If the prior year’s noise were significantly less than the limit, the airport could increaseflight slots available to the airlines. But if noise exceeded the limit, the airport would reducethe number of slots available to those airlines with the noisiest operations (City of LongBeach Ordinance, Chapter 16.43). At LHR, the noise quotas apply to nighttime operations,between 11:30 PM and 6 AM (Wit et al. (2003)). Airlines with quieter aircraft may operatemore nighttime flights than those with noisier aircraft.

5Cumulative noise is computed using a formula that can be approximated by L = n+10 log(f).

6Note that the consumer compares frequencies for the two duopoly airlines along with thefares they charge in making a choice between them, without considering the exact timing ofindividual flights. While this approach may not be fully accurate for individual consumers,it could apply to business travelers, who cannot predict their travel times and thus purchaserefundable full-fare tickets on the airline with the best frequency/fare combination, tickets

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that allow them to board the next flight upon arriving at the airport. Alternatively, themodel could capture the choice setting of a corporate travel department, which must sign anexclusive contract with a particular airline for transporting its employees. The travel depart-ment cares about the average schedule delay for the company employees, while also seekinglow fares. It signs an exclusive contract with the airline providing the best combination ofthese features.

7While a represents the gain from traveling on airline 1 relative to airline 2, the above conditionrequires measurement of the absolute gain from traveling on 1. For an easy transitionbetween the relative and absolute cases, it is assumed that when a > 0, the absolute gain fromusing airline 1 equals a itself while the absolute gain from airline 2 equals zero. Conversely,when a < 0, the absolute gain from airline 1 equals 0 while the absolute gain from airline 2equals a itself.

8Using the inequality b < p2 + γ/f2, which holds by assumption, to eliminate b in (6) andusing a > a, it follows that (3) is satisfied for all these consumers, indicating that airline 1is also preferred to airline 2.

9As seen below, the duopoly fare and frequency solutions are independent of b.

10Girvin’s model does not rely on the present brand-loyalty framework, but instead followsthe approach of Brueckner (2004), where brand loyalty is absent but the travel benefit bvaries across consumers. This alternate approach nevertheless yields a traffic solution withthe form of (7).

11This requirement reduces to (4γ/α2f31 )(p1 − τ )− (1/α)2(ε/L + γ/f2

1 )2 > 0.

12Since abatement cost per flight equals εs/n, abatement cost per seat is ε/n = εf/L

13Recall that for the duopoly case to be relevant, the inequality b ≥ p + γ/f must hold.Substituting the p and f solutions from (11) and (12) yields a lower bound for b in terms ofthe parameters of the model. An analogous lower bound arises in the other cases consideredbelow.

14Application of the envelope theorem to the profit expression in (8) shows that airline 1’sprofit is increasing in L holding f2 fixed, indicating that profit falls as regulation is tightenedconditional on f2. But the equilibrium profit of each airline, found by substituting the sym-metric solutions from (11) and (12) into the profit function is decreasing in L, indicating thattighter regulation is instead beneficial. This conclusion is counterintuitive, but it evidentlyarises from the competitive interaction between the airlines.

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15Although the second-order for the second-stage p1 choice is satisfied, the second-order con-dition for the first-stage f1 choice requires f1

3 < αγ/3θ2, which is assumed to hold.

16One change, however, concerns the profit impact of a tighter noise constraint. The L-derivative of the first-stage profit function can be shown to equal zero, so that the previousprofit impact of the noise limit vanishes (see footnote 14).

17See Girvin (2006b) for details of the derivation. The second-order conditions under simul-taneous choice require f1 > γ/2α, or 2α2/γ > θ, while the second-order condition for thefirst-stage choice of f1 in the sequential case requires f1 > γ/3α, or 3α2/γ > θ.

18Airline profit is now independent of n under both simultaneous and sequential choice.

19As mentioned earlier, Nero and Black (2000) analyze the effect of airline taxes in a relatedmodel where noise per aircraft is fixed and taxes are levied on a per-flight basis, while Hsuand Lin (2005) analyze the effect of noise taxes in a complex simulation-based network model.

20See Girvin (2006b) for details.

21While noise taxation and cumulative noise regulation are equivalent in the model, a tax maybe superior in the case where the airlines have different characteristics (cost per seat forexample). In this case, the optimal cumulative noise limits would differ between the carriers,while a single noise tax would presumably be desirable (see the welfare analysis below).

22To show ∂f/∂ε < 0, note that (25) can be rewritten as

√γ

8θL2(L/ε)2 + 4L2(L/ε)= t.

Since L is increasing in ε, it follows that L/ε must be decreasing in ε to maintain this equalityas ε rises, implying ∂f/∂ε < 0 from (23). To show ∂p/∂ε > 0, the term ε/n in the pricesolution can be rewritten as 2Lt and the previous results for L used.

23For a cost-benefit analysis of the FAA noise regulation, see Morrison, Winston and Watson(1999).

24Empirical estimates of noise damage are generated through hedonic housing price models.See McMillen (2004) for a recent example and Nelson (2003) and Schipper, Nijkamp andRietveld (1998) for up-to-date surveys.

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25It should be noted that the airport itself, which may earn profits from its operations, isnot considered in the welfare analysis. Such profits would arise from unmodeled chargeslevied on the airlines, or from noise taxes under a tax regime. However, since airport profitsrepresent a transfer from the airlines to the airport, they would cancel in a welfare functionthat equals the sum of airline and airport profits and consumer surplus. As a result, theinterests of the airport can be ignored in the welfare analysis. Airport behavior, however,plays a key role in other types of models; for representative studies, see Morrison (1987),Oum and Zhang (1990), and Zhang and Zhang (2003).

26With the second-best values of both f and L lower than the first-best values, a comparisonof the n levels (given by f/L ratio) is inconclusive.

27Since W and none of the sequential f solutions depend on α, the value of this parameterneed not be specified. Recalling that satisfaction of the airline second-order conditions inthe first stage of sequential case is not guaranteed, it can then be assumed that α takes avalue such that these conditions hold.

28The full set of numerical results is available in Girvin (2006b). In contrast to Proposition7, numerical examples in Girvin (2006a) show that, in the monopoly case, cumulative noiseregulation is slightly superior to per-aircraft regulation everywhere in her chosen parameterspace. This finding, of course, does not establish a general result.

29These aircraft include the Boeing 777, 737-700 to -900, and 717, along with the AirbusA320/321, A330 and A340.

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